Jasonbradley
Home

(x-4)(y-7)=0 Y-5/x+y-9=2

4 min read Jun 17, 2024
(x-4)(y-7)=0 Y-5/x+y-9=2

Solving a System of Equations: (x-4)(y-7)=0 and y-5/x+y-9=2

This article will guide you through the process of solving the system of equations:

(1) (x-4)(y-7) = 0 (2) y - 5/x + y - 9 = 2

Understanding the First Equation

The first equation, (x-4)(y-7) = 0, represents a product of two factors equaling zero. This means either (x-4) = 0 or (y-7) = 0, or both. Let's break down the possibilities:

  • Case 1: x - 4 = 0 This leads to x = 4.
  • Case 2: y - 7 = 0 This leads to y = 7.

Working with the Second Equation

The second equation, y - 5/x + y - 9 = 2, is more complex. To solve for x and y, we need to:

  1. Simplify the equation: Combine like terms and move the constant term to the right side: 2y - 5/x = 11

  2. Isolate y: Multiply both sides by x: 2xy - 5 = 11x

  3. Solve for y: Rearrange the equation to express y in terms of x: y = (11x + 5) / 2x

Combining the Solutions

Now, we need to combine the solutions we found from the first equation with the expression for y from the second equation:

  • If x = 4: Substitute x = 4 into the expression for y: y = (11 * 4 + 5) / (2 * 4) = 49 / 8

  • If y = 7: Substitute y = 7 into the expression for y: 7 = (11x + 5) / 2x 14x = 11x + 5 3x = 5 x = 5/3

Solutions

Therefore, the solutions to the system of equations are:

  • x = 4, y = 49/8
  • x = 5/3, y = 7

Verification

It's always a good idea to verify our solutions by plugging them back into the original equations:

  • For (x = 4, y = 49/8): (4-4)(49/8-7) = 0 * 0 = 0 49/8 - 5/4 + 49/8 - 9 = 2

  • For (x = 5/3, y = 7): (5/3 - 4)(7 - 7) = 0 * 0 = 0 7 - 5/(5/3) + 7 - 9 = 2

Conclusion

We successfully solved the system of equations using a combination of factoring, simplification, and substitution. The solutions (x = 4, y = 49/8) and (x = 5/3, y = 7) satisfy both equations in the system.

Related Post

Featured Posts